The Islamic holy books are the records which most Muslims believe were dictated by God to various prophets. They are the Suhuf Ibrahim (commonly the Scrolls of Abraham), the Tawrat (Torah), the Zabur (commonly the Psalms), the Injil (commonly the Gospel), and the Qur'an.

The words Tawrat and Injil are mentioned several times in the Qur'an itself. For example, following ayat has these words with Kitab, meaning book:

He has sent down upon thee the Book with the truth, confirming what was before it, and He sent down the Torah and the Gospel aforetime, as guidance to the people, and He sent down the Salvation.(Qur'an 3:3-4 Arthur Arberry translation)

He has revealed to you the Book with truth, verifying that which is before it, and He revealed the Tavrat and the Injeel aforetime, a guidance for the people, and He sent the Furqan. (Qur'an 3:3-4 Shakir translation)

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Other possible books or prophets

The Qur'an does not say that additional holy books were sent to other prophets, but one book with all prophets is mentioned in Quran, For example.: Mankind was one single nation, and Allah sent Messengers with glad tidings and warnings; and with them He sent the Book in truth, to judge between people in matters wherein they differed; (Qur'an 2:213).
It is not standard Islamic belief, but most of the Muslims believe that all holy scriptures except the Qur'an have been altered from their original forms over time, Tahrif. The Qur'an mentions other prophets or messengers like Ibraheem (Abraham), Is'haq (Isaac), Isma'il (Ishmael), Lot (Luth), Yaqoob (Jacob), Joseph (Yusof), Job (Ayub), Dawud (David), Sulayman (Solomon), Musa (Moses), Harun (Aaron), Yunus (Jonah), Elisha (Ilyasa'), Zakariya (Zachariah), Yahya (John) and Isa (Jesus) and all prophets got one Al-kitab (The Book), (Qur'an 6:89 translation)
Yousuf Ali 6:89:
"These were the men to whom We gave the Book, and authority, and prophethood: if these (their descendants) reject them, Behold! We shall entrust their charge to a new people who reject them not."

The Qur'an's relationship to earlier books

Many Muslims believe in progressive revelation, that the revelation of God (Arabic: Allah) evolved with time and different groups of people, but The Quran says that the books, from which the people are studying the concept of previously revealed holy books are not sent by Allah, Yusuf Ali 34:44: But We had not given them Books which they could study, nor sent apostles to them before thee as Warners.

Pickthal 34:44: And We have given them no scriptures which they study, nor sent We unto them, before thee, any warner.
As an example, while the Qur'an acknowledges that the Torah prohibited working on the Sabbath, the Qur'an allows working and overrules it. In Muhammad's earlier years it was revealed to him, "O People of the Book! Ye have no ground to stand upon unless ye stand fast by the Torah, (and) the Gospel." Qur'an Surah 5.68. It was believed that their conversion to Islam would begin by devoutly following the earlier holy books.